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Symmetry reduction for nonlinear partial differential equations

Workshop
Part of the Lecture Notes in Physics book series (LNP, volume 189)

Keywords

Nonlinear Evolution Equation Nonlinear Partial Differential Equation Symmetry Reduction Generic Orbit Nonlinear Sigma Model 
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References

  1. [1]
    M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Nonlinear evolution equations of physical significance. Phys. Rev. Lett. 31, 125–127 (1973).CrossRefGoogle Scholar
  2. [2]
    M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, The inverse scattering transform: Fourier analysis for nonlinear problems. Studies Appl. Math. 53, 249–315 (1974).Google Scholar
  3. [3]
    M. J. Ablowitz, A. Ramani, and H. Segur, A connection between nonlinear evolution equations and ordinary differential equations of P-type, I and II. J. Math. Phys. 21, 715–721 and 1006–1015 (1980).CrossRefGoogle Scholar
  4. [4]
    M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform. SIAM, Philadelphia, 1981.Google Scholar
  5. [5]
    M. J. Ablowitz and H. Segur, Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38, 1103–1106 (1977).CrossRefGoogle Scholar
  6. [6]
    W. F. Ames, Nonlinear Ordinary Differential Equations in Transport Processes. Academic Press, New York, 1968.Google Scholar
  7. [7]
    W. F. Ames, in Nonlinear Equations in Abstract Spaces. V. Lakshmikantham ed., Academic Press, New York, 1978, pp. 43–66.Google Scholar
  8. [8]
    W. F. Ames, Nonlinear Partial Differential Equations in Engineering, Vols. I and II. Academic Press, New York, 1965 and 1972.Google Scholar
  9. [9]
    R. L. Anderson, A nonlinear superposition principle admitted by coupled Riccati equations of the projective type. Lett. Math. Phys. 4, 1–7 (1980).CrossRefGoogle Scholar
  10. [10]
    R. L. Anderson, A. O. Barut, and R. Raczka, Bäcklund transformations and new solutions of nonlinear wave equations in four-dimensional space-time. Lett. Math. Phys. 3, 351–358 (1979).CrossRefGoogle Scholar
  11. [11]
    R. L. Anderson, J. Harnad and P. Winternitz, Group-theoretical approach to superposition rules for systems of Riccati equations. Lett. Math. Phys. 5, 143–148 (1981).CrossRefGoogle Scholar
  12. [12]
    R. L. Anderson, J. Harnad and P. Winternitz, Systems of ordinary differential equations with nonlinear superposition principles. Physica 4D, 164–182 (1982).Google Scholar
  13. [13]
    R. L. Anderson and N. H. Ibragimov, Lie-Bäcklund Transformations in Applications. SIAM, Philadelphia (1979).Google Scholar
  14. [14]
    P. Appel and E. Lacour, Principes de la Théorie des Fonctions Elliptiques et Applications. Gauthier-Villars, Paris, 1922.Google Scholar
  15. [15]
    A. V. Bäcklund, Zur Theorie der Partiellen Differentialgleichungen erster Ordnung. Math. Ann. 17, 285–328 (1880).CrossRefGoogle Scholar
  16. [16]
    A. Barone, F. Esposito, C. J. Magee, and A. C. Scott, Theory and applications of the sine-Gordon equation. Nuovo Cimento 1, 227–267 (1971).Google Scholar
  17. [17]
    J. Beckers, J. Harnad, M. Perroud, and P. Winternitz, Tensor fields invariant under subgroups of the conformal group. J. Math. Phys. 19, 2126–2153 (1978).Google Scholar
  18. [18]
    J. Beckers, J. Harnad, M. Perroud, and P. Winternitz, Subgroups of the euclidean group and symmetry breaking in nonrelativistic quantum mechanics. J. Math. Phys. 18 72–83 (1977).Google Scholar
  19. [19]
    L. Bianchi, Ricerche sulle superficie a curvatura constante e sulle elicoidi. Ann. Scuola Norm. Sup. Pisa 2, 285 (1879).Google Scholar
  20. [20]
    G. W. Bluman and J. D. Cole, The general similarity solution of the heat equation. J. Math. Mech. 18, 1025–1042 (1969).Google Scholar
  21. [21]
    G. W. Bluman and J. D. Cole, Similarity Methods for Differential Equations. Applied Mathematical Sciences #13, Springer Verlag Lecture Notes in Mathematics, 1974.Google Scholar
  22. [22]
    G. W. Bluman and S. Kumei, On the remarkable nonlinear diffusion equation \(\tfrac{\partial }{{\partial x}}\left[ {a\left( {u + b} \right)^{ - 2} \tfrac{{\partial u}}{{\partial x}}} \right] - \tfrac{{\partial u}}{{\partial t}} = 0\). J. Math. Phys. 21, 1019–1023 (1980).Google Scholar
  23. [23]
    M. Boiti and F. Pempinelli, Similarity solutions of the Korteweg-de Vries equation. Nuovo Cimento 51B, 70–78 (1979).Google Scholar
  24. [24]
    M. Boiti and F. Pempinelli, Nonlinear Schrödinger equation, Bäcklund transformations and Painlevé transcendents. Nuovo Cimento 51B, 40–58 (1980)Google Scholar
  25. [25]
    M. Boiti and F. Pempinelli, Similarity solutions and Bäcklund transformations of the Boussinesq equation. Nuovo Cimento 56B, 148–156 (1980).Google Scholar
  26. [26]
    C. P. Boyer, E. G. Kalnins, and W. Miller Jr., Symmetry and the separation of variables for the Hamilton-Jacobi equation W t2-W z2-W y2 = 0. J. Math. Phys. 19, 200–211 (1978).Google Scholar
  27. [27]
    C. P. Boyer, E. G. Kalnins, and W. Miller Jr., Separable coordinates for four-dimensional Riemannian spaces. Commun. Math. Phys. 59, 285–302 (1978).Google Scholar
  28. [28]
    C. P. Boyer, E. G. Kalnins, and P. Winternitz, Completely integrable relativistic hamiltonian systems and separation of variables in hermitian hyperbolic spaces. Preprint CRMA-1104 (1982), to appear in J. Math. Phys. 24 (1983).Google Scholar
  29. [29]
    C. P. Boyer, E. G. Kalnins, and P. Winternitz, Separation of variables for the Hamilton-Jacobi equation on complex projective spaces. Preprint CRMA-1064 (1981), submitted to SIAM J. Math. Anal.Google Scholar
  30. [30]
    C. P. Boyer, R. T. Sharp, and P. Winternitz, Symmetry breaking interactions for the time dependent Schrödinger equation. J. Math. Phys. 17, 1439–1451 (1976).CrossRefGoogle Scholar
  31. [31]
    R. K. Bullough and P. J. Caudrey (eds.), Solitons. Springer Verlag, 1980.Google Scholar
  32. [32]
    F. Calogero and A. Degasperis, Spectral Transform and Solitons. North Holland, Amsterdam, 1982.Google Scholar
  33. [33](a)
    L.-L. Chau, Integrability of self dual Yang-Mills equations and the role of the Kac-Moody algebra. Lectures in these Proceedings.Google Scholar
  34. [33](b)
    L.-L. Chau, Bianchi-Bäcklund transformations, conservation laws, and linearization of various field theories. In The High Energy Limit, A. Zichichi ed., Plenum Publ. Corp. 183, pp. 249–279.Google Scholar
  35. [34]
    P. L. Christiansen and P. S. Lomdahl, Numerical study of 2+1-dimensional sine-Gordon solitons. Physics 2D, 482–494 (1981).Google Scholar
  36. [35]
    J. Clairin, Sur les transformations de Bäcklund. Ann. Sci. Ecole Norm. Sup. 3, Supplément, 1–63 (1902).Google Scholar
  37. [36]
    W. J. Coles, Matrix Riccati differential equations. SIAM J. Appl. Math. 13, 627–634 (1965).Google Scholar
  38. [37]
    J. Corones, Solitons and simple pseudopotentials. J. Math. Phys. 17, 756–759 (1976).Google Scholar
  39. [38]
    J. Corones and F. J. Testa, Pseudopotentials and their applications. In Bäcklund Transformations, Proceedings, Nashville Tenn., 1974, R. M. Miura, ed. Springer Verlag, 1976.Google Scholar
  40. [39]
    H. T. Davis, Introduction to Nonlinear Differential Equations. Dover, New York, 1962.Google Scholar
  41. [40]
    M. C. Delfour, E. B. Lee, and A. Manitius, F-reduction of the operator Riccati equation for hereditary differential systems. Automatics 14, 385–395 (1978).Google Scholar
  42. [41]
    C. Dubois, Règles de superposition pur les équations de Riccati rnatricielles générées sous U(n,n) et O(n,n). M. Sc. Thesis, Université de Montréal, 1982.Google Scholar
  43. [42]
    H. Eichenherr, SU(n)-invariant nonlinear σ-models. Nucl. Phys. B146, 215–223 (1978).Google Scholar
  44. [43] (a)
    H. Eichenherr and M. Forger, On the dual symmetry of the nonlinear sigma models. Nucl. Phys. B155, 381–393 (1979)Google Scholar
  45. [43] (b)
    ib., More about nonlinear sigma models on symmetric spaces. Nucl. Phys.B164, 528–535 (1980)CrossRefGoogle Scholar
  46. [43] (c)
    ib., Higher local conservation laws for nonlinear sigma models on symmetric spaces. Commun. Math. Phys.82, 227–255 (1981)Google Scholar
  47. [44]
    L. P. Eisenhart, Continuous Groups of Transformations. Dover, New York, 1961.Google Scholar
  48. [45] (a)
    F. B. Estabrook, Some old and new techniques for the practical use of exterior differential forms. In Bäcklund Transformations, Proceedings, Nashville Tenn., 1974, R. M. Miura, ed. Springer Verlag, 1976Google Scholar
  49. [45] (b)
    F. B. Estabrook and H. D. Wahlquist, Prolongation structures of nonlinear evolution equations. J. Math. Phys. 17, 1293–1297 (1976).CrossRefGoogle Scholar
  50. [46]
    M. Flato, G. Pinczon, and J. Simon, Nonlinear representations of Lie groups. Ann. Scient. Ecole Norm. Sup. 4 t. 10, 405–418 (1977).Google Scholar
  51. [47] (a)
    M. Flato and J. Simon, Nonlinear equations and covariance. Lett. Math. Phys. 2, 155–160 (1977)CrossRefGoogle Scholar
  52. [47] (b)
    ib., Yang-Mills equations are formally linearizable. Lett. Math. Phys.3, 279–283 (1979).CrossRefGoogle Scholar
  53. [48]
    M. Flato and J. Simon, Linearization of relativistic nonlinear wave equations. J. Math. Phys. 21, 913–917 (1980).CrossRefGoogle Scholar
  54. [49]
    M. Flato and J. Simon, On a linearization program of nonlinear field equations. Phys. Lett. 94B, 518–522 (1980).Google Scholar
  55. [50]
    A. S. Fokas, Generalized symmetries and constants of the motion of evolution equations. Lett. Math. Phys. 3, 467–473 (1979).CrossRefGoogle Scholar
  56. [51]
    A. S. Fokas, A symmetry approach to exactly solvable evolution equations. J. Math. Phys. 21, 1318–1325 (1980).CrossRefGoogle Scholar
  57. [52]
    A. S. Fokas and M. J. Ablowitz, On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23, 2033–2043 (1982).CrossRefGoogle Scholar
  58. [53]
    A. S. Fokas and Y. C. Yortsos, The transformation parameters of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento 30, 539–544 (1981).Google Scholar
  59. [54]
    A. S. Fokas and R. L. Anderson, Group theoretical nature of Bäcklund transformations. Lett. Math. Phys. 3, 117–126 (1979).CrossRefGoogle Scholar
  60. [55]
    B. Gambier, Sur les équations différentielles du second ordre et du premier degré dent l'inrégrale générale est à points critiques fixes. Acta Math. 33, 1–55 (1910).Google Scholar
  61. [56]
    C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett. 19, 1095–1097 (1967).CrossRefGoogle Scholar
  62. [57]
    P. G. Glockner and M. C. Singh (eds.), Symmetry, Similarity, and Group Theoretic Methods in Mechanics. University of Calgary, Calgary, 1974.Google Scholar
  63. [58]
    M. Golubitsky, Primitive actions and maximal subgroups of Lie groups. J. Diff.Geom. 7, 175–191 (1972).Google Scholar
  64. [59]
    B. Grammaticos, B. Dorizzi, and R. Padjen, Painlevé property and integrals of motion for the Henón-Heiles system. Phys. Lett. 89A, 111–113 (1982).Google Scholar
  65. [60]
    A. M. Grundland, J. Harnad, and P. Winternitz, Solutions of the multidimensional sine-Gordon equation obtained by symmetry reduction. Kinam 4, 333–344 (1982).Google Scholar
  66. [61]
    A. M. Grundland, J. Harnad, and P. Winternitz, Symmetry reduction for nonlinear relativistically invariant equations. Preprint CRMA-1162 (1983), submitted to J. Math. Phys..Google Scholar
  67. [62]
    J. Harnad, Y. Saint-Aubin, and S. Shnider, Superposition of solutions to Bäcklund transformations for the SU(n) principal sigma model. Preprint CRMA-1074 (1982).Google Scholar
  68. [63]
    J. Harnad, S. Shnider, and Y. Saint-Aubin, Quadratic pseudopotentials for GL(n,C) principal sigma models. Preprint CRMA-1075 (1982).Google Scholar
  69. [64]
    J. Harnad, S. Shnider, and J. Tafel, Group actions on principal bundles and dimensional reduction. Lett. Math. Phys. 4, 107–113 (1980).CrossRefGoogle Scholar
  70. [65]
    J. Harnad, S. Shnider, and L. Vinet, Solutions to Yang-Mills equations on M 4 invariant under subgroups of O(4,2). In Complex Manifold Techniques in Theoretical Physics. Pitman Research Notes in Mathematics 32, 219–230 (1979).Google Scholar
  71. [66]
    J. Harnad, S. Shnider, and L. Vinet, The Yang-Mills system in compactified Minkowski space; invariance conditions and SU(2) invariant solutions. J. Math. Phys. 20, 931–942 (1979).CrossRefGoogle Scholar
  72. [67]
    J. Harnad, S. Shnider, and L. Vinet, Group actions on principal bundles and invariance conditions for gauge fields. J. Math. Phys. 21, 2719–2724 (1980).CrossRefGoogle Scholar
  73. [68]
    J. Harnad and L. Vinet, On the U(2) invariant solutions to Yang-Mills equations in compactified Minkowski space. Phys. Lett. 6B, 589–592 (1978).Google Scholar
  74. [69]
    J. Harnad and P. Winternitz, Pseudopotentials and Lie symmetries for the generalized nonlinear Schrödinger equation. J. Math. Phys. 23, 517–525 (1982).CrossRefGoogle Scholar
  75. [70]
    J. Harnad, P. Winternitz, and R. L. Anderson, Superposition principles for matrix Riccati equations. Preprint CRMA-1024 (1981), to appear in J. Math. Phys. 24, (1983)Google Scholar
  76. [71]
    B. K. Harrison and F. B. Estabrook, Geometric approach to invariance groups and solutions of partial differential equations. J. Math. Phys. 12, 653–666 (1971).CrossRefGoogle Scholar
  77. [72]
    S. Helgason, Differential Geometry, Lie groups, and Symmetric Spaces. Academic Press, New York, 1978.Google Scholar
  78. [73]
    E. Hille, Ordinary Differential Equations in the Complex Domain. John Wiley, 1976.Google Scholar
  79. [74]
    N. Kh. Ibragimov, Gruppovyîe Svoîatva Nîekotorykh Differentsîalnykh Urav ieniî, (Group Theoretical Properties of some Differential Equations). Nauka, Novosibirsk, 1967.Google Scholar
  80. [75]
    E. L. Ince, Ordinary Differential Equations. Dover, New York, 1956.Google Scholar
  81. [76]
    H. Joos, Zur Darstellungstheorie der inhomogenen Lorentzgruppe als Grundlage quantenmechanischer Kinematik. Fortschr. d. Phys. 10, 65–146 (1962).Google Scholar
  82. [77]
    E. G. Kalnins and W. Miller Jr., Killing tensors and variable separation for Hamilton-Jacobi and Helmholtz equations. SIAM J. Math. Anal. 11, 1011–1026 (1980).CrossRefGoogle Scholar
  83. [78]
    E. G. Kalnins and W. Miller Jr., Killing tensors and nonorthogonal variable separation for Hamilton-Jacobi equations. SIAM J. Math. Anal. 12, 617–629 (1981).CrossRefGoogle Scholar
  84. [79]
    E. G. Kalnins, W. Miller Jr., and P. Winternitz, The group O(4), separation of variables and the hydrogen atom. SIAM J. Appl. Math. 30, 630–664 (1976).CrossRefGoogle Scholar
  85. [80]
    D. J. Kaup, The Estabrook-Wahlquist method with examples of application. Physics 1D, 391–411 (1980).Google Scholar
  86. [81]
    K. K. Kobayashi and M. Izutsu, Exact solution of the n-dimensional sine-Gordon equation. J. Phys. Soc. Japan 41, 1091–1092 (1976).Google Scholar
  87. [82]
    K. K. Kobayashi and T. Nagano, On filtered Lie algebras and geometric structures, I and I. J. Math. Mech. 13, 875–908 (1964), and 14, 513–521 (1965).Google Scholar
  88. [83] (a)
    Y. Kosmann-Schwarzbach, Sur les transformations de similitude des equations aux dérivées partielles. C. R. Acad. Sc. Paris 287A, 953–956 (1978)Google Scholar
  89. [83] (b)
    ib., Generalized symmetries of nonlinear partial differential equations. Lett. Math. Phys.3, 395–404 (1979).CrossRefGoogle Scholar
  90. [84]
    V. Kučera, A review of the matrix Riccati equation. Kybernetika 9, 92–61 (1973).Google Scholar
  91. [85]
    S. Kumei and G. W. Bluman, When nonlinear differential equations are equivalent to linear differential equations. SIAM J. Appl. Math. 42, 1157–1174 (1982).CrossRefGoogle Scholar
  92. [86]
    G. L. Lamb Jr., Bäcklund transformations for certain nonlinear evolution equations. J. Math. Phys. 15, 2157–2165 (1974).CrossRefGoogle Scholar
  93. [87]
    G. L. Lamb Jr., Elements of Soliton Theory. John Wiley, 1980.Google Scholar
  94. [88]
    P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467–490 (1968).Google Scholar
  95. [89]
    S. Lie, Allgemeine Untersuchungen über Differentialgleichungen die eine continuirliche endliche Gruppe gestatten. Math. Ann. 25, 71–151 (1885).CrossRefGoogle Scholar
  96. [90]
    S. Lie, Vorlesungen über differentialgleichungen mit bekannten infinitesimalen Transformationen. Teubner, Leipzig, 1891); reprinted by Chelsea Publ. Co., New York, 1967.Google Scholar
  97. [91]
    S. Lie and F. Engel, Theorie der Transformationgruppen. Teubner, Leipzig, 1888 (Vol. 1); 1890 (Vol. 2); 1893 (Vol. 3); reprinted by Chelsea Publ. Co., New York, 1967.Google Scholar
  98. [92]
    S. Lie and G. Scheffers, Vorlesungen über continuierliche Gruppen mit geometriechen und anderen Anwend. Teubner, Leipzig, 1893; reprinted by Chelsea Publ. Co., New York, 1967.Google Scholar
  99. [93]
    G. Leibrandt, New exact solutions of the classical sine-Gordon equation in 2+1 and 3+1 dimensions. Phys. Rev. Lett. 41, 435–438 (1978).CrossRefGoogle Scholar
  100. [94]
    A. I. Malcev, Foundations of Linear Algebra. W. H. Freeman, San Francisco, 1963.Google Scholar
  101. [95]
    J. B. McLeod and P. J. Olver, The connection between partial differential equations soluble by inverse scattering, and ordinary differential equations of Painlevé type. MRC Report 2135, University of Wisconsin, 1980.Google Scholar
  102. [96]
    W. Miller Jr., Symmetry and the Separation of Variables. Addison-Wesley, Reading Mass., 1977.Google Scholar
  103. [97]
    W. Miller Jr., The Technique of Variable Separation for Partial Differential Equations. Lectures in these Proceedings.Google Scholar
  104. [98]
    W. Miller Jr., J. Patera, and P. Winternitz, Subgroups of Lie groups and the separation of variables. J. Math. Phys. 22, 251–260 (1981).CrossRefGoogle Scholar
  105. [99]
    H. C. Morris, Prolongation structures and generalized inverse scattering problems. J. Math. Phys. 17, 1867–1869 (1976).CrossRefGoogle Scholar
  106. [100]
    H. C. Morris, Prolongation structures and nonlinear evolution equations in two spatial dimensions. J. Math. Phys. 17, 1870–1872 (1976).CrossRefGoogle Scholar
  107. [101]
    H. C. Morris, Prolongation structures and nonlinear evolution equations in two spatial dimensions. I. A generalized nonlinear Schrödinger equation. J. Math. Phys. 18, 285–288 (1977).CrossRefGoogle Scholar
  108. [102]
    T. D. Newton, The inhomogeneous Lorentz group. In Theory of Groups in Classical and Quantum Physics, T. Kahan, ed. Am. Elsavier, 1966.Google Scholar
  109. [103]
    T. Ochiai, Classification of the finite nonlinear primitive Lie algebras. Trans. AMS 124, 313–322 (1966).Google Scholar
  110. [104]
    A. T. Ogielski, M. K. Prasad, A. Sinha, and L.-L. Chau Wang, Bäcklund transformations and local conservation laws for principal chiral fields. Phys. Lett. 91B, 387–391 (1980).Google Scholar
  111. [105]
    P. J. Olver, Symmetry groups and group invariant solutions of partial differential equations. J. Diff. Geometry 14, 497–542 (1979).Google Scholar
  112. [106]
    L. V. Ovsannikov, Grupovoî Analiz Differentsialnikh Uravnîeniî, (Group Theoretical Analysis of Differential Equations). Nauka, Moscow, 1978.Google Scholar
  113. [107]
    P. Painlevé, Sur les équations différentielles du second ordre et d'ordre supérieur dont l'integrale générale est uniforme. Acta Math. 25, 1–85 (1902).Google Scholar
  114. [108]
    J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhaus, Subgroups of the similitude group of three-dimensional Minkowski space. Can. J. Phys. 54, 986–994 (1976).Google Scholar
  115. [109]
    J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhaus, Subgroups of the Poincare group and their invariants. J. Math. Phys. 17, 1439–1451 (1976).CrossRefGoogle Scholar
  116. [110]
    J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhaus, Continuous subgroups of the fundamental groups of physics. III. The de Sitter groups. J. Math. Phys. 18, 2259–2288 (1977).CrossRefGoogle Scholar
  117. [111]
    J. Patera and P. Winternitz, A new basis for the representations of the rotation group. Lamb and Henn polynomials. J. Math. Phys. 14, 1130–1139 (1973).CrossRefGoogle Scholar
  118. [112] (a)
    J. Patera, P. Winternitz, and H. Zassenhaus, Continuous subgroups of the fundamental groups of physics. I. general Method and the Poincaré group. J. Math. Phys. 16, 1597–1614 (1975); ib., II. The similitude group. J. Math. Phys. 16, 1615–1624 (1975).CrossRefGoogle Scholar
  119. [113]
    F. A. E. Pirani, D. C. Robinson, and W. F. Shadwick, Local Jet Bundle Formulation of Bäcklund Transformations. D. Reidel, Dordrecht, 1979.Google Scholar
  120. [114]
    K. Polhmeyer, Integrable hamiltonian systems and interactions through quadratic constraints. Commun. Math. Phys. 46, 207–221 (1976).CrossRefGoogle Scholar
  121. [115]
    Z. Popowicz, Painlevé ping-pong P 3-P b. Workshop contribution in these Proceedings.Google Scholar
  122. [116]
    D. Rand, Etude Numérique des Règles de Superposition pour les Equations Matricielles de Riccati. M. Sc. Thesis, Université de Montréal, 1982.Google Scholar
  123. [117]
    D. Rand and P. Winternitz, Nonlinear superposition principles: a new numerical method for solving matrix Riccati equations. Preprint CRMA-1124 (1982).Google Scholar
  124. [118]
    W. T. Reid, Riccati Differential Equations. Academic Press, New York, 1972.Google Scholar
  125. [119]
    Y. Saint-Aubin, Bäcklund transformations and soliton type solutions for σ-models with values in real Grassmannian spaces. Preprint CRMA-1106 (1982). To appear in Lett. Math. Phys. Google Scholar
  126. [120]
    W. F. Shadwick, The Bäcklund problem for the equation \(\tfrac{{\partial ^2 z}}{{\partial x^1 \partial x^2 }} = f(z)\). J. Math. Phys. 19, 2312–2317 (1978).CrossRefGoogle Scholar
  127. [121]
    A. C. Scott, F. Y. F. Chu, and D. W. McLaughlin, The soliton: a new concept in applied science. Proc. IEEE 61, 1443–1483 (1973).Google Scholar
  128. [122]
    S. Shnider and P. Winternitz, Classification of systems of nonlinear ordinary differential equations with superposition principles. Preprint CRMA-1164, 1983.Google Scholar
  129. [123]
    M. Sorine, Sur l'équation de Riccati stationnaire associée an probléme de contrôle d'un système parabolique. C. R. Acad. Sc. Paris 287A, 445–448 (1978).Google Scholar
  130. [124]
    M. Sorine and P. Winternitz, Superposition laws for nonlinear equations arising in optimal control theory. To be published.Google Scholar
  131. [125]
    E. Tailin, Analytic linearization, hamiltonian formalism, and infinite sequences of constants of motion for the Burgers equation. Phys. Rev. Lett. 47, 1425–1428 (1981).CrossRefGoogle Scholar
  132. [126]
    G. Temple, Lectures on topics in nonlinear differential equations. Report 1415, David Taylor Model Basin, Carderock Md., 1960.Google Scholar
  133. [127]
    Tu Gue-Zhang, The Lie algebra of the invariance group of the KdV, MKdV, or Burgers equation. Lett. Math. Phys. 3, 387–393 (1979).CrossRefGoogle Scholar
  134. [128]
    M. E. Vessiot, Sur les sysèmes d'équations différentielles du premier ordre qui ont des systèmes fondamentaux d'intégrales. Ann. sc. Ecole Norm. sup. 10, 53 (1983).Google Scholar
  135. [129] (a)
    H. De Wahlquist and F. B. Estabrook, Prolongation structures of nonlinear evolution equations. J. Math. Phys. 16 1–7 (1975)CrossRefGoogle Scholar
  136. [129] (b)
    ib., Bäcklund transformation for solutions of the Korteweg-de Vries equation. Phys. Rev. Lett.31, 1386–1390 (1973)CrossRefGoogle Scholar
  137. [130]
    G. B. Whitham, Linear and Nonlinear Waves. John Wiley, 1974.Google Scholar
  138. [131]
    G. B. Whitham, Comments on some recent multisoliton solutions. J. Phys. A12, L1–L3 (1979).Google Scholar
  139. [132]
    E. P. Wigner, On unitary representations of the inhomogeneous Lorentz group. Ann. Math. 40, 149–204 (1939).MathSciNetGoogle Scholar
  140. [133]
    J. C. Willems, Least square stationary optimal control and the algebraic Riccati equation. IEEE Trans. Autom. Control AC16, 621–634 (1971).CrossRefGoogle Scholar

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